isometric deformation of a plane
Final:

As far as I know your best shot would be with Curve modifier. It will not be perfectly even though but I don't think it's possible to preserve 1:1 edge length with any modifier.



Steps:




Add Curve > Bezier Curve.
Rotate it 90 degrees on X axis.
Select Plane and add Curve Modifier.
Choose BezierCurve.
Move a Plane on X axis.